If |x ^ 2 - 6x + 8| = 1 then possible values of x
(a) 3 + sqrt(2)
(b) 3 plus/minus sqrt(8)
(c) 3, 3 - sqrt(2)
(d) None of these

The given equation is:

\[
|x^2 - 6x + 8| = 1
\]

To solve this, we need to consider two cases because of the absolute value.

### Case 1: \( x^2 - 6x + 8 = 1 \)
First, solve the equation without the absolute value:

\[
x^2 - 6x + 8 = 1
\]

Subtract 1 from both sides:

\[
x^2 - 6x + 7 = 0
\]

Now, solve this quadratic equation using the quadratic formula:

\[
x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(7)}}{2(1)}
\]

Simplifying further:

\[
x = \frac{6 \pm \sqrt{36 - 28}}{2} = \frac{6 \pm \sqrt{8}}{2} = \frac{6 \pm 2\sqrt{2}}{2} = 3 \pm \sqrt{2}
\]

So, the solutions from this case are:

\[
x = 3 + \sqrt{2} \quad \text{and} \quad x = 3 - \sqrt{2}
\]

### Case 2: \( x^2 - 6x + 8 = -1 \)
Now consider the case where the expression inside the absolute value equals \(-1\):

\[
x^2 - 6x + 8 = -1
\]

Add 1 to both sides:

\[
x^2 - 6x + 9 = 0
\]

This is a perfect square:

\[
(x - 3)^2 = 0
\]

So, \(x = 3\).

### Final Answer:
The possible values of \(x\) are \(3\), \(3 + \sqrt{2}\), and \(3 - \sqrt{2}\).

So, the correct answer is:

**(c) 3, 3 - \(\sqrt{2}\)**

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Would you like more details on how the quadratic formula is applied, or any other questions?

Here are some related questions you might find helpful:
1. How do you solve absolute value equations?
2. How do you complete the square in quadratic equations?
3. What is the geometric interpretation of quadratic equations?
4. How do you determine the nature of the roots using the discriminant?
5. How do you solve quadratic equations graphically?

**Tip:** When solving absolute value equations, always remember to consider both the positive and negative cases of the expression inside the absolute value.

Learn how to solve the absolute value equation |x^2 - 6x + 8| = 1 with detailed steps and explanations. This problem involves quadratic equations and absolute value equations, covering the quadratic formula and its application. Suitable for high school students.

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